Weyl group action on weight zero Mirkovic-Vilonen basis and equivariant multiplicities

We state a conjecture about the Weyl group action coming from Geometric Satake on zero-weight spaces in terms of equivariant multiplicities of Mirkovic-Vilonen cycles. We prove it for small coweights in type A. In this case, using work of Braverman, Gaitsgory and Vybornov, we show that the Mirkovic-Vilonen basis agrees with the Springer basis. We rephrase this in terms of equivariant multiplicities using work of Joseph and Hotta. We also have analogous results for Ginzburg's Lagrangian construction of sl(n) representations. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The study discusses the Weyl group action on zero-weight spaces and proposes a conjecture regarding equivariant multiplicities of Mirkovic-Vilonen cycles. It is proven to hold for small coweights in type A, with the Mirkovic-Vilonen basis found to agree with the Springer basis. The results are reformulated in terms of equivariant multiplicities using the work of Joseph and Hotta, and similar results are obtained for Ginzburg's Lagrangian construction of sl(n) representations.